What are the divisors of 1080?

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, 1080

There is a total of 32 positive divisors.
The sum of these divisors is 3600.
The arithmetic mean is 112.5.

24 even divisors

2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 40, 54, 60, 72, 90, 108, 120, 180, 216, 270, 360, 540, 1080

8 odd divisors

1, 3, 5, 9, 15, 27, 45, 135

How to compute the divisors of 1080?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 1080 by each of the numbers from 1 to 1080 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

1080 / 1 = 1080 (the remainder is 0, so 1 is a divisor of 1080)
1080 / 2 = 540 (the remainder is 0, so 2 is a divisor of 1080)
1080 / 3 = 360 (the remainder is 0, so 3 is a divisor of 1080)
...
1080 / 1079 = 1.0009267840593 (the remainder is 1, so 1079 is not a divisor of 1080)
1080 / 1080 = 1 (the remainder is 0, so 1080 is a divisor of 1080)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 1080 (i.e. 32.86335345031). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

1080 / 1 = 1080 (the remainder is 0, so 1 and 1080 are divisors of 1080)
1080 / 2 = 540 (the remainder is 0, so 2 and 540 are divisors of 1080)
1080 / 3 = 360 (the remainder is 0, so 3 and 360 are divisors of 1080)
...
1080 / 31 = 34.838709677419 (the remainder is 26, so 31 is not a divisor of 1080)
1080 / 32 = 33.75 (the remainder is 24, so 32 is not a divisor of 1080)